Twisted conjugacy classes in twisted Chevalley groups

Let G be a group and phi be an automorphism of G. Two elements x,y is an element of G are said to be phi-twisted conjugate if y = gx phi(g)(-1) for some g is an element of G. We say that a group G has the R-infinity-property if the number of phi-twisted conjugacy classes is infinite for every automorphism phi of G. In this paper, we prove that twisted Chevalley groups over a field k of characteristic zero have the R-infinity-property as well as the S-infinity-property if k has finite transcendence degree over Q or Aut(k) is periodic.